A friend of mine really liked that option, don't care for it myself but was usually an option back on the 360 on most things we played. Have not kept up with the modern world though so no idea there. Not sure if there is an official name for such a control scheme either -- I know older Halo titles call it "Legacy", and that seems to be a fairly common term in my token search. I will stick that in the title for you.
Anyway if it is going to be a problem and you are going to go with a Switch (go where the games are at and all that) then it should be possible to modify a controller fairly easily to swap the relevant axes of the sticks -- most sticks will use the same encoding methods between sticks (whether resistive or encoder based) because there is no point in using multiple stick types in a device normally. Find the relevant wires coming out of them and cut traces and throw wires (it should be 4 wires at most, and possibly just two if they are referenced off the same thing (likely ground but I have been surprised in the past so check).
It will take marginally more effort to make it a switch so you can revert when you feel like it but still nothing major.
While such a thing makes more sense than some game specific button mods I see people do here if you don't want to sacrifice a pro controller then there are apparently third party controllers costing less (often far less) that are actually worth a damn these days.
There may well be a console hack or software mod option as well but the former will need a hacked machine and I am not sure people are playing with remapping right now, software ones then get done on a game by game basis and possibly worse than that (think minigame by minigame, or at least multiple approaches needed for a single game).
Also when I read the title I first thought you were going for pre WASD FPS controls.
Rotflmao 
